Basic Question About General Relativity? I'm a layman that loves Physics. I'm also horrible at math. Having said that I have many, many questions in regards to physics and General Relativity in-particular. I will try to keep my question(s) as clear as I can. 
In relation to Gravity, Galaxies and The Universe can someone please explain the following:
If the force of gravity within a given galaxy is keeping said galaxy together, as well as "pulling" all objects within it closer (towards the center and/or the Black Hole at the center), how does General Relativity account for the fact that all galaxies are racing "away" from each other, and accelerating faster and faster as they do so? 
It seems, (to me at least), that this is a "paradox" of some sort. How can it be that the bodies within a given galaxy are being drawn closer to one another over time due to gravity and yet that whole galaxy is moving away from all the other galaxies in the universe that should, by the laws of G.R., be heading towards one another as well. 
I understand a little bit about expansion, the big bang etc. However, this question/paradox has always interested me. Again as a layman it is hard for me to grasp the mathematical reasoning behind all of this, but I am however, good at conceptualizing these phenomena when laid out in a more analogous or geometric fashion.
 A: Earth is a center of attraction by gravity.  Everything falls down toward the ground, or more precisely the center of the Earth.  So how is it we can throw a ball upward?   The initial velocity is upward, and the force of gravity acts to change it to downward, but only by a certain amount per second.  9.8 meters per second, per second.    
It is the same with cosmic expansion - all the galaxies were "thrown upward" in a sense, given huge initial velocities apart from each other.  Well, no, it's a little stranger than that. We say space itself is expanding.  But never mind - there is in fact gravity attracting galaxies together, but it can act only so much, whatever kilometer/second per million years. 
A big question among astronomers decades ago was whether the gravity is strong enough, or the initial velocities slow enough, for them to all come crashing together billions of years from now, or not.  It is like asking if a ball thrown upward, or rocket if that's what you happen to have handy, is going faster than escape velocity.  We know now: they won't be pulled back together.  We also know about a mysterious repulsive force apparently acting on a huge scale, the "Dark Energy".   (We need a better name for it!)   
Within a galaxy, the forces of gravity don't change strength (not enough to matter) and the stars go merrily about each other and orbit about the central black hole (if there is one) pretty much following ordinary orbital mechanics.  It's like the molecules in the ball thrown upward, sticking together due to strong forces at close range, not minding how the whole thing is moving in a greater realm.
A: 
It seems, (to me at least), that this is a "paradox" of some sort.

For me, at least, the key idea that resolves this "paradox" is that it is only on truly vast scales, e.g., roughly 3 billion light years on a side and larger, that the universe is approximately homogeneous and isotropic.
When the equations of GR are solved for a homogeneous isotropic spacetime, the solution is a space that is either expanding or contracting and, of course, we have evidence that our universe is expanding.
If, on all scales, this homogeneous and isotropic condition were true, this solution would hold on all scales.
However, on the smaller scale of galaxy clusters, galaxies, solar systems, etc., the homogeneous and isotropic assumption does not hold.  On these scales, matter is not uniformly distributed but is instead "clumpy".
On these smaller scales, we have gravitationally bound systems of planets and moons, stars and planets, galaxy and stars, galaxies and other galaxies etc. but this is also in accord with GR.
In others words, though the very large scale and smaller scale behaviour of the universe are quite different, there is no paradox.
A: I find @DarenW's analogy somewhat confusing, so I'll propose a slightly different answer. The repulsive "force" that causes the expansion is significant on very large scales. Galaxies (also stars, planets and humans) have average density large enough to ignore it (or, to put it differently, they clump together faster than the expansion of Universe pulls them apart). If the Universe continues its accelerated expansion, galaxies, stars, planets and humans (if there are any at that point in time) will be eventually pulled apart as well.
I'd also like to provide a quote from the set of lectures "Quantum computing since Democritus" by Scott Aaranson (which I highly recommend). They are mostly dedicated to computational complexity, but in one of the last lectures he talks about cosmology:

So what is this cosmological constant? Basically, a kind of anti-gravity. It's something that causes two given points in spacetime to recede away from each other at an exponential rate. What's the obvious problem with that? As the Woody Allen character's mother told him, "Brooklyn is not expanding." If this expansion is such an important force in the universe, why doesn't it matter within our own planet or galaxy? Because on the scale that we live, there are other forces like gravity that are constantly counteracting the expansion. Imagine two magnets on the surface of a slowly-expanding balloon: even though the balloon is expanding, the magnets still stick together. It's only on the scale of the entire universe that the cosmological constant is able to win over gravity.
You can talk about this in terms of the scale factor of the universe. Let's measure the time t since the beginning of the universe in the rest frame of the cosmic background radiation (the usual trick). How "big" is the universe as a function of $t$? Or to put it more carefully, given two test points, how has the distance between them changed as a function of time? The hypothesis behind inflation is that at the very beginning–at the Big Bang–there's this enormous exponential growth for a few Planck times. Following that, you've got some expansion, but also have gravity trying to pull the universe together. It works out there that the scale factor increases as $t^{2/3}$. Ten billion years after the Big Bang, when life is first starting to form on Earth, the cosmological constant starts winning out over gravity. After this, it's just exponential all the way, like in the very beginning but not as fast.

